1=.999999...?

[UWHabs]

Most of it comes down to the fact that math defines two numbers to be equal if they are essentially "close enough" to one another that there can't possibly be any number in between the two.

And I think almost everyone would agree that if .999... and 1 were different numbers, there can't be a number between the 2 of them. But by the standard math, by definition that ends up meaning that they are equal. So by that, I might be willing to concede that they are "different but equal", but by the standard math that is taught, they are most certainly equal.

 

[Timsup2nothin]

Most of it comes down to the fact that math defines two numbers to be equal if they are essentially "close enough" to one another that there can't possibly be any number in between the two.

And I think almost everyone would agree that if .999... and 1 were different numbers, there can't be a number between the 2 of them. But by the standard math, by definition that ends up meaning that they are equal. So by that, I might be willing to concede that they are "different but equal", but by the standard math that is taught, they are most certainly equal.

Taught where? Can't be another number between them means they are equal? When did that happen? So a number is equal to "the next number," whatever that might mean, which is then equal to "the next number", etc etc etc and presto facto dipso impacto ALL NUMBERS ARE EQUAL.

That's why math defines two numbers to be equal if they are equal, not if they are "well, close enough."

The premises here just get better and better.

By the way there is no such thing as two numbers with no number between them.

 

[Borachio]

Oh? What's the integer between 1 and 2, then?

/pedantry

 

[Hehehe]

Forgive me for I did not read the entire thread, but how is there still discussion about this? It's already been proven that 0.999... = 1. As counter intuitive as it may seem to some people, you can represent 0.999 as an infinite series (0.9+0.09+0.009+...) which, when summed up, comes out to exactly 1.

 

[UWHabs]

Taught where? Can't be another number between them means they are equal? When did that happen? So a number is equal to "the next number," whatever that might mean, which is then equal to "the next number", etc etc etc and presto facto dipso impacto ALL NUMBERS ARE EQUAL.

That's why math defines two numbers to be equal if they are equal, not if they are "well, close enough."

The premises here just get better and better.

By the way there is no such thing as two numbers with no number between them.

Sorry, I meant in relation to limits of infinite series (the whole delta-epsilon arguments). .9+.09+.009+... is not necessarily equal to 1, but it becomes so close to one that it is indistinguishable. Depending on how you deal with real numbers, actually deriving a formal way to declare two numbers as equal is not a trivial thing. If you're defining real numbers as Cauchy sequences, .999... = 1 essentially because they are both constructed from infinite sequences that converge to the same number (.9, .99, .999, ... has the same limit as the sequence 1, 1, 1, ...)

 

[Atticus]

There's been plenty of times when people with less education than I said things I dismissed because it was outside what I had learned that was related to the subject at hand, only to find out later I was wrong and they were not. I was too into my categories of interpretation that I didn't consider other ones.

Has that ever happened to you?

In maths? Not that I'd recall. I've made plenty of mistakes corrected by people with less education, but don't remember that a person with substantially less education would have made a relevant point that I would have dismissed.

If someone makes a comment on maths that I disagree with, I ask for his/her justification and present my own, at least that's the principle, sometimes I won't bother. Maths is self contained that way, if you know the definitions, you can get by with reasoning.

However, I've several times mistakenly thought that I know some things better than I do, due to superficial studying or mere arrogance. I could dig some cases out from this forum too, but won't bother right now.

So, you and Terx say that education may be hindrance on understanding the opposition's view here, but have you considered this: maybe your lack of education prevents you from understanding the case made against your argument? If you have thought about it, what convinced you that that's not what happening? How did you become sure that you really understood all the arguments presented here?

Some of the things said here aren't easily digested. For example the axioms of the real numbers that I linked to and the epsilon-delta method of proving things, people spend time understanding it. They ask questions, do exercises, get helped... Even after the first course it may take some time to digest the whole picture.

In this thread it looks like many posters just scanned through those posts and deemed them irrelevant. I wish I would have had students who understood not only the axioms of the real numbers, but also all the ramifications of them and the epsilon-delta method from a single reading of a forum post. Unfortunately, I've never come across with one.

 

[onejayhawk]

Sorry, I meant in relation to limits of infinite series (the whole delta-epsilon arguments). .9+.09+.009+... is not necessarily equal to 1, but it becomes so close to one that it is indistinguishable. Depending on how you deal with real numbers, actually deriving a formal way to declare two numbers as equal is not a trivial thing. If you're defining real numbers as Cauchy sequences, .999... = 1 essentially because they are both constructed from infinite sequences that converge to the same number (.9, .99, .999, ... has the same limit as the sequence 1, 1, 1, ...)

Everyone agrees that the limit is the same. However, every term of the sequence is an element of (0,1), but the limit is not. How can a term equal the limit of the sequence. It cannot. 0.999... is such a term, ergo =/= 1.

J

 

[UWHabs]

Everyone agrees that the limit is the same. However, every term of the sequence is an element of (0,1), but the limit is not. How can a term equal the limit of the sequence. It cannot. 0.999... is such a term, ergo =/= 1.

J

Every term of the sequence is less than one, but the limit is still one. In the same way that the limit of 1/2, 1/3, 1/4, ... is zero, even though every term is non-zero. Or though of another way, the sequence 1, 2, 3, ... has a limit of infinity, but every term in it is finite.

However where I think you're not picking up is that the term we've been calling .999... IS the limit of that series. If it has an infinite number of digits, it's obviously not a term in the series at any finite spot, so it must be the limit as the series is clearly constructed that way.

 

[Leifmk]

0.999... is such a term, ergo =/= 1.

J

Ah, there's the rub. No, 0.999... is not some nth term of the sequence for some finite number n, it is in fact the same as the limit of the sequence.

 

[Borachio]

Everyone agrees that the limit is the same. However, every term of the sequence is an element of (0,1), but the limit is not. How can a term equal the limit of the sequence. It cannot. 0.999... is such a term, ergo =/= 1.

J

I think a lot of the difficulty comes from thinking that infinity is just some arbitrarily large number, N, when it really isn't. I don't think it's a number at all, tbh.

 

[GenMarshall]

Personally, if I'm not obsessing with accuracy. I typically round it up to 1. Though if I see something like .993~, then I drop everything after the 3. Though that's just me and dealing with anything with money and cents.

 

[onejayhawk]

Ah, there's the rub. No, 0.999... is not some nth term of the sequence for some finite number n, it is in fact the same as the limit of the sequence.

Yes it is. That is EXACTLY what it is. Which is why it is =/= 1, which is the same as the limit of the sequence.

J

 

[Souron]

I think a lot of the difficulty comes from thinking that infinity is just some arbitrarily large number, N, when it really isn't. I don't think it's a number at all, tbh.

What's a number anyway?

I think it's best to think of infinity as a number in some contexts, and not in others. It's a choice of how you define your number system.

 

[Souron]

Forgive me for I did not read the entire thread, but how is there still discussion about this? It's already been proven that 0.999... = 1. As counter intuitive as it may seem to some people, you can represent 0.999 as an infinite series (0.9+0.09+0.009+...) which, when summed up, comes out to exactly 1.

An infinite series is defined as the limit of a particular function at infinity. Specifically here it is limit of the the sum from 1 to i of 9/(10^i) converges at 1 as i approaches infinity. In general the limit of a function at x is not necessarily equal to the limit of that function at x. So this is a special rule that applies only to infinite series. The question of 0.999... = 1 therefore hinges on if you choose accept this axiom. And it is an axiom; you cannot prove it. But it is a definition of infinite series that allows us to do useful arithmetic, so it is generally accepted. But if you don't accept the axiom, then 0.999... does not equal 1.

 

[Folket]

This thread is a dark abyss of human insanity. I mostly feel like trolling it but I will probably not.

To say that 0.99.. is not 1 because your family can't write that number down is nonsense. Instead if having your family write nines forever imagine yourself writing infinite numbers of nines in one second. That way you get the number in finite time instead of at the end of infinite time which does not exists.

Now you have the number and can confirm that it is the same number as 1 with different notation.

Taught where? Can't be another number between them means they are equal? When did that happen? So a number is equal to "the next number," whatever that might mean, which is then equal to "the next number", etc etc etc and presto facto dipso impacto ALL NUMBERS ARE EQUAL.

You seem to misunderstand how real works. There exists no "the next number", that is what was said. Any number smaller or larger has infinite number between them.

Also just because there are things like infinitesimal and non standard analysis does not mean that in that there is a useful set of axioms where 0.99... =/= 1. Even in NSA 0.99.. = 1.

 

[onejayhawk]

This thread is a dark abyss of human insanity. I mostly feel like trolling it but I will probably not.

To say that 0.99.. is not 1 because your family can't write that number down is nonsense. Instead if having your family write nines forever imagine yourself writing infinite numbers of nines in one second. That way you get the number in finite time instead of at the end of infinite time which does not exists.

Now you have the number and can confirm that it is the same number as 1 with different notation.

You seem to misunderstand how real works. There exists no "the next number", that is what was said. Any number smaller or larger has infinite number between them.

Also just because there are things like infinitesimal and non standard analysis does not mean that in that there is a useful set of axioms where 0.99... =/= 1. Even in NSA 0.99.. = 1.

It is not enough to say "infinite number between them." The real numbers are not countable. There is an uncountable number of numbers between any two numbers. Therein is the rub. How do you distinguish between what is unsupported in a countable infinity, but is supported in an uncountable infinity? Or vice versa.

The much referred to Cauchy sequence, .9, .99, .999, ... is a countable infinity of terms, all of which are strictly less than 1, ie less than and not equal. Yet, the limit of the sequence =1. Further, all elements of this sequence are elements of (0,1). The limit is not.

How do you justify the leap from being an element of a set to being an element to the compliment of the set?

J

 

[Atticus]

Why should it be justified? It's just a thing that happens, there's no reason to think it couldn't happen.

You can prove however that the limit has to be in the closure of the set where the sequence lies. In this case, [0,1].

(And btw., it's complement).

 

[Borachio]

What's a number anyway?

I think it's best to think of infinity as a number in some contexts, and not in others. It's a choice of how you define your number system.

Good question. And not one that had occurred to me before, so in desperation I looked up the rough and ready definition:

an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations.

Can infinity ever be used in this way? I don't think, atm, that it can.

 

[Atticus]

You can add infinity and minus infinity to the reals by defining
a+infinity = infinity for all real a
infinity + infinty = infinity
infinity - infinity = not defined
etc. The set you get is usually called "the extended reals" or something like that and is written R with a bar over it (\bar{\mathbb{R}}, if you read LaTex).

That can be useful for example in measure theory, which examines "sizes" of sets. Some sets are infinitely large, i.e. their measure is infinite, and it's much easier to treat that like a special number than to handle it another way.

A different thing would be notational convenience, where the infinity isn't actually treated like a number. For example, you want to write
lim a_k = infinity
for a sequence like (k²).
That's however just a shorthand for saying that the sequence grows out of all bounds, i.e.
For every M in reals there is a k_M such that a_k > M whenever k > k_M.
Notice that this definition isn't the same as that of the ordinary limit. The sequence doesn't get any nearer to the infinity (if you choose to have one, or in a colloquial sense). No matter how far you go in it, you're still infinitely far away from "the limit".

 

[Terxpahseyton]

That way you get the number in finite time instead of at the end of infinite time which does not exists.

That it does not exist is the point you $%/&/&(&()/&(%

Anyway, I am out of this thread It is a purely philsophical topic from my POV and really not worth the massive massive wall of arrogant ignorance having me get angry over and over Not all were like that, of course! Not always.

I'll post a last reply to hoplitejoe because he raised a point I wasn't aware of and found interesting, but that is all. Now I need to go to bed, I just couldn't resist taking a small peak. No more, thank you!